..ATOMIC ENERGY OF CANADA LIMITEDrAJ Power Projects, Sheridan Park, Ontario Lecture 4

ATOMIC ENERGY OF CANADA LIMITEDPower Projects

NUCLEAR POWER SYMPOSIUM

LECTURE NO.4: REACTOR PHYSICS

by

E. Critoph

1. "SPLITTING THE ATOM": A SOURCE OF ENERGY

In 1937-8 when I was a young boy, 8 or 9 years old, living in B. C.,I had a friend who was considerably older attending high school. Hewas a constant source of new information, and I can still rememberhim telling me several times that if a way to "split the atom" werediscovered it would make an enormous store of energy available.I, of course, found this very interesting but at the same time extremelybaffling. I couldn't understand (a) how "splitting the atom" couldproduce energy, (b) how anyone knew that "splitting the atom" wouldproduce energy if they didn't know how to do it and hence presumablyhad never done it, and (c) why someone didn't just go ahead and "splitthe atom" in a brute force way (maybe with a small sharp axe).

In case any of you have had similar questions and have not beenfortunate enough to have them entirely resolved, I would like to takethe first few minutes of my talk to recall the situation in 1937-8, andtry to show you the answers to this type of question.

1. 1 Equivalence of Mass and Energy

Well before 1937-8 Einstein's famous formula for the eqUivalence ofmass and energy, which was first proposed in 1903 as a consequenceof his Special Theory of Relativity, was generally accepted.

In this formula, E = mC2, if the mass, m, is in grams, and thevelocity of light, C, is in cm/sec, then the energy, E, will be in ergs.

Therefore, since the velocity of light is 3 x 1010 em/sec,

1 Ib (454 g) of mass is eqUivalent to 4. 1 x 1023 ergsor 4. 1 x 1016 watt-secor 1. 14 x 1010 kWhor 4.75 x 105 MWd

2

To put this in perspective: to get 1. 14 x 1010 kWh of heat frombituminous coal requires the burning of 1,500,000 tons.

1. 2 Mass Defect

To see how this fits in with the possibility of acquiring energy by"splitting the atom" we must now consider a phenomenon called themass defect.

One might think offhand that, since any atom is composed entirely ofneutrons, protons and electrons, the mass of the atom would be thesum of the masses of the neutrons, protons and electron present inthe atom. However, careful measurements of atomic masses showthat they are always less than this and the difference is called themass defect for that atom.

The mass defect arises from the fact that there are strong nuclearforces holding the protons and neutrons in a nucleus of an atomtogether. The mass defect represents the energy which would bereleased in the process of assembling the particular atom from therequisite number of protons, neutrons and electrons. The sameamount of energy would have to be supplied to break up an atom intoits constituent parts (protons, neutrons and electrons). Thc energyequivalent of the mass defect is appropriately called the bindingenergy of the particular nuclide.

1. 3 Binding Energy Per Nucleon

If the binding energy of a nuclide is divided by the number of nucleons(i. e., protons plus neutrons), the result is the binding energy pernucleon. Figure 1 is a plot of the binding energy per nucleon as afunction of mass number (i. e., number of protons plus neutrons in thenucleus).

1. 4 Energy From "Splitting The Atom"

Let us examine this curve of binding energy per nucleon vs massnumber a little more closely and see what it tells us. Notice that thecurve starts at quite low values for light elements, climbs to a maxi­mum for mass numbers in the range 50-100 and then decreases slowlyfor further increases in mass number. It is this last slow decreasewhich explains why energy can be obtained by "splitting the atom".

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BBINDINGENERGY 6

PERNUCLEON

4(MeV)

2

3

/' ---~f '-......I

oo so 150

MASS No (A)

260

Figure 1 Binding Energy Per Nucleon Vs. A

As an example let us calculate the energy released if we take a poundof material with mass number 250 and split each nuclide into twonuclides of mass number 125.

Total number of nucleons (protons + neutrons):

~ 454 x 6 x 1023

~ 2. 72 x 1026 nucleons

From Figure 1 the binding energy per nucleon goes from about 7.3MeV for mass number 250 to 8.4 MeV for mass number 125. There­fore, the total increase in binding energy is:

(S.4 - 7.3)(2.72 x 1026) MeV

= 3 x 1026 MeV.

(The MeV is a unit of energy:

1 MeV::: 1. 6 x 10-6 ergs

::: 4.5 x 10-20 kWh)

Therefore, the total energy released in "splitting the atoms" in 1 poundof mass number 250 material is:

(3 x 1026)(4.5 x 10-20) ;::: 1. 35 x 107 kWh

(or the equivalent of burning 1750 tons of coal).

Note that in doing this we have only converted a little over 0.1% of theoriginal mass into energy.

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1. 5 Fusion

Note also in Figure 1 that if we were to combine the nuclides of a verylight element together to form nuclides with twice the number ofnucleons, much more energy would be released per nucleon than isobtained by splitting heavy atoms. This process is referred to asfusion and has, of course, great potential for power production. Thereare still very difficult problems to surmount in using fusion on a largescale, and since this is not the subject of this talk, I will say no moreabout it.

We have now answered the questions of how "splitting the atom" canlead to energy release, and how people realized this was so beforeknowing how to do it. We will now go on to a discussion of a discoverywhich made it feasible to use this process as a basis for commercialpower production.

2. FISSION

In 1939 a way to split the nucleus of certain heavy isotopes was dis­covered by Hahn and Strassmann. They found that when these isotopeswere bombarded by neutrons (which were discovered in 1932 byChadwick and about which we will say more later), some of the nucleidivided into two more or less equal fragments with, of course, anassociated release of energy. This process of division was calledfission.

Fission can also occur spontaneously in a number of materials,particularly in uranium and plutonium isotopes.

However, this is a rare occurrence as can be seen from the fact thatfor 238U (which is the uranium isotope which undergoes spontaneousfission most easily) it takes 8 x 1015 years for half the nuclides in alarge sample to fission spontaneously. The spontaneous fission half­lives of a number of isotopes are given in Figure 2.

There are three very important characteristics of fission, namely:

(i) Two nuclides, of approximately equal mass, are produced andthese are usually radioactive. These are called fission productsor fission fragments. Figure 3 shows the distribution of fissionproducts obtained when 235U is fissioned by neutrons.

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ISOTOPE S.F. 1/2 LIFE

233U ~3 x 1017 VRS.

234U 1.6 x 1016 VRS.

235U 2 x 10 17 VRS.

236U 2 x 1016 VRS.

238U 8 x 1015 VRS.

239PU 5.5 x 1015 VRS.

240Pu 1.2 x 1011 YRS.

242pu 6.6 x 1010 VRS.

Figure 2 Spontaneous Fission Half-Lives

10 r---=,.-----.-------::=~--_,

.1<f<0...J .01IU

>=.001

.0001

.0000180 100 120

MASS NUMBER

140 160

Figure 3 Fission Product Yields

6

(ii) Energy is released for the reasons we have discussed. Mostof this energy is in the form of kinetic energy of fission frag­ments but some is also in the form of 13 -and y -ray energy,neutrinos, and neutron kinetic energy. Figure 4 is a tabulationof the distribution of energy release among these forms, forfission of 235U caused by neutrons.

ENERGYFORM PER

FISSION (MeV)

I. K.E. FISSION FRAG. 167.0

2. K.E. NEUTRONS 4.8

3. PROMPT /"S 7.2

4. F.P. "Y'S 6.1

5. F.P.j3·S 7.4

6. NEUTRINOS 10.4

202.9

Figure 4 Distribution of Energy Release from 235U Fission

(iii) Typically 2 or 3 neutrons are produced by the fission process.Figure 5 lists the average number of neutrons produced perfission in severai fissile isotopes when fission is caused by"thermal" neutrons (the term "thermal" neutron will beexplained later).

ISOTOPE V

233U 2.49

235u 2.43

239pu 2.87

241Pu 2.97

Figure 5 Average Number of Neutrons per Fission (v)

7

The bulk (> 99%) of the neutrons produced as a result of fission are"promptltj that is, they are produced during the actual fission process.A small fraction of the neutrons are produced later by the decay offission products. These are referred to as "delayed" neutrons and weshall see later that these are very important when we come to thinkingof devices for commercial power production. Figure 6 shows thepercentage of neutrons released as delayed neutrons for neutron­induced fission of several isotopes. Six groups of delayed neutronshave been identified, each having a characteristic half-life associatedwith their production. Figure 7 shows the distribution among thesegroups of the delayed neutrons from 235U fission as well as the half­lives associated with the groups.

The prompt neutrons produced in fission are quite energetic. Figure 8shows the distribution of neutron energies from thermal neutroninduced fission of 235U. The average kinetic energy is about 2 MeVand the most probable kinetic energy around 1 MeV. (An energy of1 MeV corresponds to a velocity of about 107 m/s). :Delayed neutronshave definite energies associated with them depending on which groupthey belong to. These energies are lower than the average energy forprompt neutrons, being in the neighbourhood of a few tenths of anMeV.

ISOTOPE% NEUTRONS

DELAYED

233U .265

235U .650

239Pu .212

Figure 6 %Delayed Neutrons from Neutron-Induced Fission

8

DELAYED HALF %OFALLNEUTRON LIFE NEUTRONS

GROUP (SEC) IN GROUP

1 55.7 .021

2 22.7 .142

3 6.2 .128

4 2.3 .257

5 .61 .075

6 .23 .027

Figure 7 Distribution of Delayed Neutronsfrom Neutron-Induced Fission of 235U

N(E)

to 2

E (MeV)

3 4

Figure 8 Fission Neutron Energy Spectrum

3.

4.

9

THE CHAIN REACTION

In providing a method of "splitting the atom", nature has been kinderto us than was strictly necessary. The fission process not only solvesthe problem of how to "split the atom" but does so in a way whichmakes it easy to achieve large scale power production by means of achain reaction.

The fission process, which is initiated by a neutron, itself producesmore than 2 neutrons. Therefore, as long as we have a neutron totrigger the first fission we are automatically supplied with moreneutrons t.o produce further fissions which, in turn, produce furtherneutrons and so on. Provided we are not too wasteful of the neutrons(i. e., by losing them in other ways than by causing fission) theprocess can continue until most of the fissile material has been con­sumed. Hence we can easily derive most of the stored energy fromthe fissile material without supplying any external energy (sinceenough neutrons to start the process are available naturally).

NEUTRONS

By now I am sure most of you are convinced that it would be arelatively simple task to design a reactor (i. e., a device for usingfission to produce neutrons and energy by means of a chain reaction)and we have not yet started to consider the main body of reactorphysics. I do not want to disillusion you, since reactor design is inprinciple simple, but there are a few details which I have ratherglossed over which must be understood before one could successfullydesign a good reactor. Since neutrons are the key agent in making areactor work, let us start by considering the properties of neutronsand the ways in which neutron interact with matter.

The neutron has no electrical charge and therefore a neutron travellingthrough a material does not interact with the electrons and nucleithrough their electric fields, but only by "direct collision" with thenuclei. Since nuclei are very small compared to atoms, neutronsgenerally travel relatively long distances (usually measured in ems)between such interactions.

4. 1 Neutron Reactions With Materials

When a neutron does collide with a nucleus there are three mainreactions which can take place: scattering, capture or fission.

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In a scattering reaction the neutron essentially bounces off the nucleus,changing its direction of motion. There are two kinds of scattering,elastic and inelastic. In elastic scattering the neutron and nucleusobey the classical laws of collisions between rigid spheres whereas ininelastic scattering there is an additional energy loss. Capture meansjust what it says - the neutron collides with the nucleus and combineswith it to form a new nucleus. Usually Sand y -rays are emittedfrom the nucleus after a neutron capture. We have already discussedthe fission reaction fairly thoroughly.

4.2 Neutron Cross Sections

The mean distance a neutron travels through a material before under­going a givcn typc of rcaction, or thc mean frce path for the reaction,is related to the neutron cross section of the material for that reaction.Neutron cross sections have the dimensions of area and are usuallyquoted in barns (1 barn = 10-24 cm2). The relationship is as follows:

1No

where1,

No

the mean free path,the number of nuclei per unit volume,neutron cross section.

(NO) is often denoted by l: and is called the macroscopic cross sectionfor the reaction. It has the dimensions of inverse length and is theprobability per unit length of neutron travel of the reaction occurring(in that material). NeutrOn cross sections have been compiled in manypublications, the best known of which is probably BNL-325. *

For example, iron (Fe) has the following neutron cross sections forneutrons with an energy of .025 eV:

Capture:Scattering:Fission:Total

2.62 barns11 barnso barns13.62 barns

Since iron has a density of 7.86 g/cm3 and a gram atomic weight of55.85 g, the number of nuclei/cm3 is:

6.02 x 102355.85 (7.86) = 8.48xI022/cm3

* BNL = Brookhaven National Laboratory

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(6.02 x 1023 is Avagadro's number or the number of atoms in a gramatomic weight.)

Therefore, the macroscopic cross sections are:

L. c 0.222 cm-1 [i. e. (8.48 x 1022)(2.62 x 10-24)]

L. s 0.933 cm-1

L.f 0

L. t 1. 15 cm-1

and the mean free paths are:

t c 4.5 cm

is 1. 07 em

tf CD

tt 0.87 em

(The total mean free path. 1. t. is the mean distance between collisionsof any type. )

Neutron cross sections are functions of neutron energy. Capture andfission cross sections usually have the general behaviour shown inFigure 9. At low neutron energies the cross section is inverselyproportional to the neutron velocity. Going up in energy we enter the"resonance region" in which the cross section fluctuates widely. Athigher energies it becomes quite small and decreasing.

Scattering cross sections also vary with energy but usually far lessthan fission and capture cross sections.

4. 3 Neutron Flux and Reaction Rates

In a nuclear reactor producing significant power, as you can wellimagine, there are neutrons flying about in all directions. If we couldtake an instantaneous picture we could count the number of neutronsper unit volume. This is called the neutron density denoted by p(see Figure 10). If now we were able to take a time exposure wecould measure the total neutron track length in a unit volume in a unittime. This is called "neutron flux", denoted by,s (see Figure 11).

t

12

1/u RESONANCEHIGH

ENERGY

'-

'"~ ~ A"'-~~

,\ I t/l~v u~

~--.001 1 1000 105

NEUTRON ENERGY (eV)

Figure 9 Typical Behaviour of Neutron Captureor Fission Cross Section With Energy

•

• •

•

Figure 10 Neutron Density, P

13

"'.-r

,,"'" fI/1.-_-"5:----­,

IL...

e-- ...,"-----7

"'--

Figure 11 Neutron Flux, s6

,s =pv

where v is the average neutron velocity.

The reaction rate per unit volume (i. e., the number of reactions takingplace per second per unit volume) for any particular neutron reactionis given by:

Reaction rate = No¢

where N is the number of nuclei per unit volume,(j is the neutron cross section of the nuclide for that

reaction (suitably averaged over the range of neutronenergies present),

and ~ is the neutron flux.

4.4 Neutron Slowing Down

We have seen that the neutrons emitted in fission are energetic, withhigh velocities. For such energetic neutrons, in most materials thescattering cross section is much higher than the fission or capturecross sections. In these materials the neutron, on the average,undergoes many scattering collisions before it is captured or causesfission. The question is "What effect do these scattering collisionshave on the neutron?" The answer is that the neutron slows down.

Since in most cases elastic scattering is the dominant process we caneasily apply the conservation laws for collisions between rigid spheresto calculate the extent of this slowing down.

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The results are fairly simple. If the neutron is scattered by a nucleuswith mass number A, and if Ei is the energy before the collision andEf the energy after the collision:

(1 : ~r Ei S Ef S E i

For example:

(i) After a collision with a deuterium nucleus (A = 2) the neutronenergy lies between the initial energy and one-ninth of theinitial energy.

(ii) After a collision with a mercury nucleus (A =200) the neutronenergy lies between the initial energy and O. 98 times theinitial energy.

The probability of any final energy between these limits is almostconstant.

There is a lower limit to the energy to which the neutron will beslowed down, which depends on the temperature of the medium. Theatoms of the medium are in motion and so the neutron eventuallyreaches a range of energy where it is just as likely to gain energy ina collision with a nucleus as lose it. When this happens the neutronis in thermal equilibrium with the medium and such neutrons arereferred to as "thermal neutrons". Such neutrons have a distributionin energy called a "Maxwellian" distribution (see Figure 12). Atroom temperature the most probable neutron velocity in such a dis­tribution is 2200 m/s corresponding to an energy of .025 eV.

The number of elastic collisions required to reduce the neutron energyfrom fission energies (~ 2 x 106 eV) to thermal energies (;;;:: .025 eV)is given by:

18.2-~-

where ~ is the average logarithmic energy decrement per collisionand is given by:

~ = 1 + (A - 1)2

1, n A - 12A A + 1

2~ A + 2/3 (for A > 10)

Figure 13 shows ~ vs A.

15

dn-n-

E-----

Figure 12 Maxwellian Distribution

1.0

.8

t .6

.4

.2

\~- ..•...-

,r\

""-~~~

2 4 6 8 10 12 14 16A

Examples:

Figure 13 ~ vs A

(i) For fission neutrons slowing down in deuterium (A = 2):

~ == O. 725 and NeOLL

- 25

(ii) For fission neutrons slowing down in carbon (A :::: 12):

~ ::: 0.158 and NeOLL

= 112

16

4.5 Neutron Migration

The other important effect of scattering collisions is that they changethe direction of motion of the neutron. On each scattering the direc­tion is changed by any amount up to 1800 and as a result the path 'traced out by the neutron is very erratic.

We already know how to calculate the mean path length of neutronsbetween different types of reactions. However, it is obvious that inorder to understand reactors in detail we must have a statisticalknowledge of the spatial distribution of the reactions arising fromneutrons born at a certain point. Therefore, we must relate ourknowledge of mean free paths to spatial distances, taking into accountthe erratic paths caused by scattering.

There are several ways of doing this which differ in accuracy anddifficulty of application. Unfortunately we do not have time to go intothese in detail, and the best I can do is to give you a vague feeling forthree of them: diffusion theory, transport theory and Monte Carlomethods.

Diffusion Theory makes two very sweeping assumptions:

(i) The probability of scattering is very much larger than thatfor capture or fission.

(ii) Scattering isotropic (i. e., the probability of the scatteredneutron going off in any direction) is constant.

With these assumptions a differential equation can be derived whichrelates the value of neutron flux at any position in a region to thematerial properties of the region (macroscopic cross sections),neutron sources in the region, and neutron fluxes and currents at theboundaries of the region. This is called the diffusion equation.

where S is the source per unitvolume per second.

Diffusion theory also relates the net neutron current at a boundary tothe shape of the flux distribution. In diffusion theory the relationshipbetween path lengths and spatial distances turns out to be:

17

r;iDiffus ion length := ~~

where ;2" is the root mean square distance travelled.

Transport theory does not make the two simplifying assumptions ofdiffusion theory and therefore gives more accurate results. On theother hand it is more difficult and time consuming to apply. Fortunatelymodifications to diffusion theory based on transport theory can be madewhich increase the accuracy without making application more difficult.Diffusion theory with transport corrections is probably the most widelyused method.

I thought Monte Carlo methods worth mentioning since they areconceptually so simple, although in practice often too time consumingto be practical. As suggested by the name, the method embodies theuse of chance. In essence the histories of many neutrons arefollowed. All decisions as to distance between reactions, type ofreactions, and change in neutron direction are made by selectingrandom numbers. Information on spatial distribution of events is thenbased on these histories.

So far we have treated neutrons slOWing down and migration separately,whereas they occur at the same time.

MUlti-group theory is usually used to combine them. The neutronsare divided into energy groups. Within any group the neutrons areassumed to obey diffusion theory with transport corrections, ortransport theory. Neutrons slowing down below the lower energy limitof any group are treated as being captured with :tespect to that groupand as a source in one of the lower energy groups.

5. A SIMPLE REACTOR

Let us now consider the simplest reactor imaginable - basically justa sphere of 235U.

5. 1 Critical Size

The first question we must answer is how big to make the sphere inorder to just sustain a chain reaction.

18

Suppose a fission takes place in the sphere producing 2.6 neutrons.These neutrons will eventually either escape from the sphere or beabsorbed (i. e., a capture or fission reaction) by the 235U, althoughthey may first undergo several scatterings. Let the probability thata neutron will not escape from the sphere before being absorbed be P.Since for fission energy neutrons the ratio of fission to absorptioncross section in 235U is O. 85, the neutrons from the original fissionwill cause (2.6 x 0.85 x P) fissions.

If the neutrons from one fission cause on the average one more fissiona chain reaction is just sustained and the reactor is said to be "critical".In our simple reactor the "criticality condition" is:

2.6 X 0.85 x P :=;: 1

or P = 0.453

The neutrons from the first fission are said to belong to one genera­tion and the neutrons from the fissions which they cause belong to thenext generation and so on.

The "infinite multiplication factor" (kco ) is the ratio of the number ofneutrons in one generation to the number in the previous generation,neglecting leakage from the system (or assuming an infinite systcm).The"effective multiplication factor" (ken) is the ratio of the numberof neutrons in one generation to the number in the previous generationtaking lcakagc into account. The criticality condition in general isgiven by kerr :::: 1.

In our simple case then:

k CD 2.6 x .85 = 2. 21,

keff ~ P :::: 2.21 P

and the criticality condition is:

keff

or P

2.21 p:::: 1,

0.453.

Obviously P is a function of the size of the sphere and provided we cancalculate this function we can find the size for whichP = • 453. Thisis the ltcritical size".

The accurate calculation of "non-leakage probability" (P) requires theuse of one of the methods mentioned under neutron migration and wedo not have time to discuss this in detail.

19

However, let us make a very crude estimate of P for the purpose ofillustration using only the knowledge we have acquired.

To do this assume:

(i) that there is no scattering, and

(ii) that the leakage probability is the same for all neutrons in thesphere independent of position.

For a neutron at the centre of the sphere the leakage probability,1 - P, is the probability that the neutron will traverse a thickness Rof 235U (where R is the radius of the sphere) without being absorbed.

To calculate this probability we observe that if n neutrons are perpen­dicularly incident on a thin slice of material dx the number of neutronsabsorbed in passing through the slice, -dn, is given by:

-dn = M dxa

in RIntegrating

dn ={ Zadxno n 0

Thereforen -ZaR

1 - Peno

For a fission spectrum the absorption cross section of 235U is 1. 65barns.

Assuming a density of 18.7 g/cm3 for 235U , the number of atoms of235v per cm3 is given by:

236.0~;510 (18.7) 4.8x1022/cm3,

Therefore Za (4.8 x 1022)(1.65 x 10~24) =.079 cm- 1

Therefore p 1 -.079R-e

The value of R for which P is 0.453 is 7.7 cm

Therefore, this crude estimate gives a sphere with a critical radiusof 7.7 cm.

20

The critical mass is:

4/3 7T(7. 7)3 (18. 7 x 10-3) == 35.8 kg

A reactor, GODrvA, which was very similar to our simple reactor wasbuilt and operated in the U. S. The main difference was that it wasmade of 93.9% 235U (plus 6.1% 238U) whereas we have assumed 100%235U• Experimentally it was found that 48. 8 kg of 235U was neededfor criticality.

The difference in U isotopic composition would lead to a small changecompared to the errors in our crude estimate due to the assumptionswe made. In fact, the reason we came as close as we did is that thetwo assumptions tend to lead to compensating errors - assuming noscattering tends to overestimate leakage whereas assuming allneutrons have the same leakage probability as those at the centretends to underestimate it.

5.2 Neutron Flux VS Power

As a matter of interest let us calculate the average neutron flux in thesphere if it is operating at a power of 100 W.

where P is the power,Ef is the energy produced/fission(NV) is the total number of 235U atoms in the sphere,

fJ f is the fission cross section,~ is the neutron flux.

Substituting;

Ef 3.2 x 10-11 watt - s per fission,

(NV) == 48~:~0 (6.02 x 1023) == 1. 25 x 1026

gives

If we solve the transport equation to obtain the flux distribution in thisreactor we find that the flux peaks in the centre and is approachingzero at the surface.

21

5.3 Control

In our simple reactor we have discussed the condition for criticality(keff == 1).

If kcff > 1 the reactor is said to be supercritical and the neutron flux.fission rate, and power will increase with time. If keff < 1 the reactoris sub-critical and the neutron flux, fission rate, and power willrlecrease with time.

Obviously in order to use a reactor we must be able to control it;making it supercritical when we want to start up or raise power, andsub-critical to shut down or lower power. We must do this bychanging the reactivity, i. e., the amount by which keff diffel:'s from l.In Canada the most common unit of reactivity is the milli-k which is10-3 in keff.

There are several ways in which we could provide a variation in keffin our simple reactor in order to permit us to go supercritical,critical, or sub-critical at will.

(i) We could make the sphere a little bigger than required forcriticality but split it in half and arrange to make the distancebetween the two hemispheres variable. Then with the hemi­spheres together the reactor would be supercritical. As thehemispheres were separated a position would be reached wherethe reactor was just critical and further separation would makeit sub-critical. This, we control the reactor by varying theleakage. This was actually the method used in GODIVA.

(ii) A second method of control would be to again make the sphere abit larger than required for criticality and then to drill a holethrough it. The sphere radius' and hole size would be chosen sothat the reactor was a bit 8upercritical with the hole empty hutsub-critical when a rod of purely absorbing material was fullyinserted in the hole. The rod would change the ratio of fissionto absorption cross section in the reactor and hence keff. Byadjusting the position of the rod in the hole we could again controlthe power of the reactor.

Similarly we could make the rod of fissile material and attain controlbut the sign of the reactivity change with direction of motion of therod would be reversed.

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5.4 Kinetics and Safety

We have seen how to control our simple reactor but we have no ideayet as to the sensitivity of the reactor to changes in reactivity.

Suppose we are using leakage control, with the rea.ctor running a.tsteady power, and suddenly we move the hemispheres closer togetherby enough to increase the reactivity by 1 mk (i. e., increase keff to1. 001).

Let us first assume that all the neutrons are prompt and see whatwould happen. The absorption mean free path of fission neutrons inpure 235U is about 12.5 em, and their average velocity is about107 m/s. Therefore the average lifetime of a neutron between birthand absorption is about 10-8 secondR. ThuR, in 1 millisecond (10- 3 s)

there are about 105 neutron generations and the power would increaseby a factor of (1. 001)105 ::::: 2.7 x 1043 .

(If keff - 1 ;::. Pex' and the neutron lifetime is denoted by £, the poweras a function of time would be given by:

(since tn (1 + x) ::::: x for x small)

tP(t) = Poe T

where T ~ the reactor period = £ /Pex

Such extreme sensitivity to changes of keff would make such a reactordangerous and very difficult to control. Fortunately it is not this bad.In the above analysis we neglected the delayed neutrons and these,although only making up about 0.7% of the neutrons from 235U fission,have a tremendous effect on the kinetics. How does this come about?

Since the delayed neutrons are emitted with half-lives of the order ofseconds, for times short compared to this, they do not reflect thesudden change in reactivity. Therefore, in the above example themultiplication factor between successive generations is initially not1. 001 but

(.007

/+ .993)(1. 001),

P Po

where plPo is the ratio of power to initial power.

23

Thus, we can see that for p/Po = 1. 17 the multiplication factor isonly 1. This means that in the first second or so the power cannotrise above 1. 17 times its initial value (compared with the previouslycalculated factor of 2. 7 x 1043 in 1 ms when we neglected delayedneutrons). In the general case, if S is the delayed neutron fractionthe initial multiplication factor between generations will be:

[s ~ + (1 - S)] (1 + P ex)

Note that if Pex > ~ the multiplication factor> 1 for any value ofp/Po, so again we would get into very fast power rises (i. e., wewould be critical on prompt neutrons alone). The fast initial risewhen Pex< S can be seen to be limited to a power ratio of

6s- Pex

After the initial fast rise the power increase settled out into anexponential rise. The period of this riSe is greatly affected by thedelayed neutrons. Figure 14 shows reactor period VB excess reactivitytaking delayed neutrons into account.

103

10

PERIOD 10.1

INSEC. 10-3

--10-5 ......

10-7

WITH NODELAYEDNEUTRONS

.0001 .001

f:). keff

.01 .1

Figure 14 Reactor Period Vs Excess Reactivity(For Simple Reactor)

24

We have only discussed the case of increased reactivity so far. Thesame type of conclusions hold for decreases in reactivity. As long asthe reactivity decrease is less than 13 the rates of change of power aregreatly affected by the delayed neutrons and are much slower than theywould be if there were no delayed neutrons.

In summary. as long as sudden reactivity changes are kept smallcompared to the delayed neutron fraction, the kinetics of the reactorare quite reasonable. Safety considerations usually require that anysudden reactivity changes larger than this be of extremely lowprobability.

6. PHYSICAL PARAMETERS AND COMPARATIVE PHYSICSOF DIFFERENT REACTOR TYPES

6.1 Changes to Simple Reactor Required for Power Production

The simple reactor which we have been talking about is not suitable foruse as a power reactor for the reasons discussed below.

(i) The power which can be extracted is too low. We would be luckyif we were able to extract a few kilowatts of heat from it and yetthe capital investment. with separated 235U costing about $12/g.would be over $0.5 million. Obviously the direction to go istowards that of heat exchanger design. We must increase thesurface area of the heat producing elements and force coolantpast these surfaces. This means the addition of coolant to thereactor core.(NOTE: It should be mentioned here that some materials,notably 238U and 232Th. are fertile, i. e., when they capture aneutron they become fissile -~ goes to 239Pu and 232Th goesto 233U. The abundance of the naturally occurring fertilematerial is several hundred times that of fissile material. Forexample, natural uranium contains roughly 137 parts of 238u toone part of 235U. Most practical power reactors use fuel con­taining some fertile material since this reduces separation costsand also makes efficient use of neutrons.)

(ii) It turns out the uranium metal fuel does not give very good per­formance in a reactor if long burnup is desired. This difficultycan be alleviated by adding alloying materials but the bettersolution is to use ceramic fuel. Currently oxide fuel is favouredalthough carbide has advantages in some cases and its use isbeing developed. Ceramic uranium fuels have a lower uraniumdensity than metallic uranium.

25

(iii) The heat producing fuel elements must be clad to prevent fissionproducts from entering the coolant and contaminating the coolantcircuits. Cladding also prevents corrosion of the fuel by thecoolant. Therefore, cladding material must be added to the core.

(iv) (Another point which will be mentioned here although it does notaffect our further discussion is that the whole reactor must beshielded with a sufficient thickness of heavy material to reduceradiation fields in the vicinity to a tolerable level. )

6.2 Effects of Adding Materials to the Core

The major changes required to make our simple reaotor suitable forpower production involve the addition of materials to the core. Whateffects will they have?

The first effect is that addition of non-fissile material (and the changeto ceramic fuel) will decrease the overall density of fissile materialin the core. For this reason alone the core will have to be madelarger.

All materials capture neutrons to a certain extent and so the additionof non-fissile material will decrease the fission to absorption ratioand hence reduce the infinite multiplication factor.

Finally all materials slow down neutrons to a certain extent and so theaddition of materials will tend to degrade the neutron spectrum (i. e. ,lower the average energy of the neutrons in the reactor).

The above situation brings us to a cross-road, as I will now attemptto show. We have already seen the general behaviour of capture andfission cross sections with neutron energy - they both tend to increaseas the neutron energy decreases. In fact, the fission cross sectionfor 235U goes from about 1. 4 barns for a fission neutron spectrum toabout 580 barns for a Maxwellian spectrum at room temperature.Ohviously there are advantages to having higher fission cross sections.

Now observe Figure 15, which shows the average number of neutronsemitted per absorption, T1 , in the fissile isotopes 235U, 239Pu, and233U. 'Yl is the average number of neutrons per fission times the ratioof fission to absorption cross section, and is a key parameter in the

I:'.:l0';)

..-----"I

,,,,\\\ \ ,,1

,'\ I___,' ....J

\\\

\,,"

233U

II I II •- I Ic I I~ I~ 31 I~c:(

ZoIX:I­~wZ

~ . " I / I / I·:L I I~ b 2'

SU b- t1{( It '"L, ,w ~ "I2 -...::enZoIX:I­~w~s::-

1071()610510410310210'10°10-'

1 _, I

10~ !

I ! I I:-~!:---L~! !

NEUTRON ENERGY (eV)

Figure 15 "Vs Neutron Energy

27

efficient use of fissile material. Figure 15 shows that 'tl is high for afission spectrum, decreases to a minimum for intermediate energies,and increases again at low energies (1. e., at thermal energies).

We can divide reactors into three general types characterized by theneutron spectrum; fast, intermediate, and thermal. In a fast reactorthe neutron energies are kept as high as feasible to take advantage ofthe high 'tl value. In a thermal reactor the neutron energies arereduced to the thermal region as completely and quickly as feasible totake advantage of the relatively good'tl value and the high fission crosssection. In between, with the neutrons at intermediate energies, liesthe intermediate reactor, and this design is to be avoided for powerproduction because of its inefficient use of neutrons.

It turns out that, to design a fast reactor, great care must be taken inthe choice of coolants. The currently favoured choice is liqUid metal(Na or NaK) although gas is a possibility. Any significant amount ofwater added to the core moves the reactor into the unfavourable inter­mediate range (unless enough is used to make a thermal reactor).This restriction in the choice of coolant, and the fact that the technologyof liquid metal coolants is difficult, explains partially why developmentof commercial fast power reactors has been so slow.

On the other hand if one opts for a thermal reactor it is advantageousto move the neutrons through intermediate energies as quickly aspossible and one purposely adds efficient slowing down materials asmoderators. It is important that the moderator have a low neutronabsorption cross section. Favoured moderators are heavy water,graphite, ordinary water, and beryllium.

Thus we see that there is no practical continuous gradation from fastreactors to thermal reactors. One must decide which design to go forand one does exactly the opposite for each with respect to moderatingmaterials.

6.4 General Features of Fast Reactors

There are several test and experimental fast reactors in operation,and a few prototype power reactors being designed and built. Com­mercial fast power reactors are probably at least a decade away.Present fast power reactor designs are all very similar and usePu02-U02 ceramic fuel and liquid sodium cooling. The fuel claddingis usually a steel alloy.

28

In the core of these reactors the fuel is in the form of elements orpencils with an outer diameter of about 1/4". The ratio of coolant tofuel plus cladding is in the neighbourhood of 1:1 and average powerdensities in the range 1/3 to 1/2 MWt per litre or about 1 MWt per kgof fissile material are anticipated. The ratio of fissile to fertilematerial is usually about 15% and the fuel must be taken to burnups inthe range 50,000 to 100,000 MWd per tonne of fissile plus fertilematerial for the concept to be economically competitive.

The core of the reactor is surrounded by a blanket containing mostlyfertile material so that the neutrons escaping from the core can beused efficiently in producing fissile material. Powers of 1000 MWeare thought to be required for competitive units.

The characteristic most often associated with fast reactors is theirability to breed, i. e., to produce more fissile material (from fertilematerial) than they consume. Thus the world's fertile resources(which are several hundred times the fissile resources) can be usedto produce power.

Some of the physics problems associated with fast reactor design haveto do with safety and fuel management. The sodium void coefficient(i. e., the effect of losing sodium coolant) can be positive, leading tosafety problems. Fuel management is very complex. Until recentlythe calculations of critical size and breeding ratio (extent of breeding)were inaccurate, but experiments on full-scale core mock-ups haveimproved the situation. The ability to accurately estimate the neutronspectrum is important.

Fast reactors are characterized by very high fuel inventories.A 1000 MWe fast power reactor will contain about 3 tonnes of fissilePu, worth at least $10/g. This high inventory contributes to the powercost and also limits the rate at which fast reactors can be introduced,even though they breed Pu.

6. 5 General Features of Thermal Reactors

In sharp contrast to fast reactors, where there is close to unanimityin choice of type, there are many types of thermal reactors alreadydeveloped or being developed. These differ mainly in choice ofmoderator - heavy water, ordinary water and graphite. However,even within the class of heavy water moderation AECL is developingthe use of three coolants - pressurized D20, boiling H 20, and organic.

29

Rather than treat all these types I will give only a brief comparisonbetween light water and heavy water thermal reactors and then concen­trate on the heavy water type.

Light water reactor cores resemble in some ways those of fastreactors, in that the U02 fuel is in the form of a more or less uniformarray of elements or pencils. These have a diameter of 0.4" to 0.6"and the ratio of coolant to fuel is about 1. 7:1 for the pressurized waterversion (PWR) and about 2. 4: 1 for the boiling light water version (BWR).The average core power densities are about 90 kW per litre for thePWR and 50 kW per litre for the BWR. Light water reactors cannotoperate with natural uranium fuel but use fuel enriched to about 2.3%in 235U and achieve burnups of about 20,000 MWd/te fissile plusfertile. The power obtained from the fissile material is about 1. 2 MWtper kg fissile material for the PWR and about O. 8 :MWt per kg fissilefor the BWR.

Since the water is heated to 300°C or so the whole core is containedin a pressure vessel. Mainly as a result of this the reactor must beshut down for refuelling.

The combination of enrichment and relatively high neutron absorptionin ordinary water leads to relatively high fuel costs. From anengineering point of view, the core is relatively compact with a highpower density. The coolant is one for which the technology is welldeveloped.

Heavy water does not slow neutrons down as quickly as light water.On the other hand it has an extremely low neutron absorption cross~ection and on balance, from the physics point of view, is a muchbetter moderator than light water. Much more moderator volume isprovided in a heavy water reactor than in a light water one; the ratioof moderator to fuel volumes being typically 20:1.

In a typical Canadian design the fuel is in the form of Zircaloy clad,U02 elements or pencils with an overall diameter of 0.6". Nineteento thirty-seven of these elements are made up into a bundle about50 em long. Nine to twelve bundles are then put end-to-end in apressure tube (3~" or 4" inner diameter). Each pressure tube has acalandria tube around it to insulate the moderator from the hot coolant.The channels are arranged in a square lattice at a pitch of 9!" to11!" and the interstitial space is filled with heavy water. The coolant,of course, flows inside the pressure tubes. A feature made possibleby this type of design is that refuelling may be done at power.

30

Heavy water reactors can use natural uranium as fuel and conversionratios (i. e., ratio of fissile atoms produced per fissile atom destroyed)of 0.83 are currently achieved. This permits burnups of about10,000 MWd/te fissile plus fertile to be achieved, about half of theenergy coming from Pu.

Although the overall power density in a heavy water core is relativelylow - typically 9 kWt/litre - the power per unit of fissile material ishigher than in either fast or light water reactors - typically approaching3 MWt/kg fissile material.

The result is a very low fuel cost, and although heavy water is expen­sive, the contribution to energy costs is low as long as losses due toleakage are kept small. The reactor core is considerably larger thana light water reactor core due to the higher moderator volume.

In the follOWing sections some of the physics aspects of heavy waterreactor design will be discussed. While the discussion will berestricted to heavy water reactors, most of the points have theircounterpart in other reactor designs.

7. PHYSICS ESTIMATES FOR PRACTICAL POWER REACTORS

Obviously many more difficulties are encountered in estimating thecharacteristics of practical power reactors than those discussed forour simple reactor. Some of these are summarized below.

(i) Variety of Materials Present:

There are several materials present in a practical power reactorrather than the single isotope in the example we treated. This,in itself, would not be too serious since we can easily takesuitable averages of properties such as cross sections. However,it does create difficulties in that properties of these materialsare often significantly different than those for 235U and introduceeffects which could be neglected in that particular case (e. g.slowing down, resonance capture). Some of these will be dis­cussed below.

(ii) Fast Fission:

In the simple reactor which we considered, fast fission in 235Uwas the main process of interest. In a natural uranium, heavywater moderated reactor this effect can be almost neglected.

31

However, fast fission in 238U becomes significant. Althou~h

23HU cannot be fissioned by thermal neutrons, above a thresholdof about 1. 4 MeV, fast neutrons can cause it to fission. Thiseffect must be taken into account.

(iii) Slowing Down:

Obviously the slowing down process must be taken specificallyinto account in calculations for any thermal reactor.

(iv) Resonance AbSOrption:

238U has very substantial capture resonances in the intermediateenergy region. Even though there is a lot of moderator, whichmoves the neutrons quickly through this region, a substantialnumber are captured. This complex effect must be treatedqUite accurately.

(v) Heterogeneity:

The heterogeneous nature of the core presents calculationaldifficulties. Since there is a fine structure in neutron fluxfrom material to material in the same neighbourhood, straighthomogenization cannot be used. Instead a detailed flux calcu­lation for a small region is usually done first and the resultsused to suitably weigh properties for use in a homogeneousreactor model.

(vi) Reactivity Coefficients:

In a power reactor there is feedback between power and reacti­vity. Macroscopic cross sections depend on material densitieswhich in turn depend on temperatures and hence powers.Resonance absorption can depend on material temperaturos(Doppler effect). The thermal neutron spectrum depends on th(~

material temperatures. The effects of power, temperature anddensity on reactivity are important from the point of view ofcontrol and safety.

(vii) Reactivity Worths:

In a reactor which uses on-power fuelling it is important to beable to estimate the effect of removing fuel from a channel.Practical reactors usually have some structural materials inthe core (such as spray nozzles, control rod guides, flux

32

detector strings, etc.) which must be accounted for. Theeffects of control and shut-off devices must be estimated.

I will mention one useful result which is derived from perturbationtheory. This is that the reactivity effects of a given absorber placedat different positions in a homogeneous reactor are proportional to theflu,.... squared at the different positions. This means that an absorberplaced at the centre of a typical reactor decreases the reactivity byfar more than the same absorber at the outSide. One of the fluxfactors accounts for the different absorption rates at the two positionsand the other for the importance of the neutrons in sustaining thechain reaction.

(viii) Non-uniformities:

In addition to being heterogeneous, most power reactors havelarge regions which differ physically. For example mostreactors have a reflector, i. e .• a region around the outsidecontaining no fuel whose purpose is to prevent as many neutronsas possible from escaping by reflecting them back into the core.

(ix) Reactor Kinetics:

Even in practical power reactors a point kinetic model is veryuseful, i. e., a model which treats changes in power, etc., asan overall effect and neglects time changes in the distributionof power. As reactors get larger it becomes more important tohave a proper spatial dynamics treatment.

8. SOME PHYSICS CHARACTERISTICS OF HEAVY WATER REACTORS

8.1 Critical Sizes and Reactivities

Using the type of lattice which has been describcd for a hcavy watcrreactor wilh freoh fuel, room temperature conditions, and D20 coolant,a cylinder 3 m in diameter and 3 m high could be made critical. Thccorc in a Bruce station reactor unit is cylindrical with a diameter ofabout 8.5 m and a length of 5,95 m. The excess reactivity associatedwith this size is about 110 mk. Of this 110 mk, about 35 mk is used inovercoming short term fission product poisons such as xenon andsamarium (see below). Another 15 mk is lost in going to operatingtemperature conditions and a few more in structural features andcontrol margins. This leaves some 50 - 60 mk in hand to look afterreactivity changes due to burnup (see below). This is enough to achievea burnup of close to 10,000 MWd/TeU from the fuel.

33

For the initial charge this excess reactivity presents some problem.If it were compensated by operating at reduced moderator level thereactor power output would have to be reduced. The methods used inCanadian reactors have included depleted uranium (t. e., uraniumwith less fissile content than natural uranium) in some of the bundlesof the initial charge, additions of horon to the moderator (which canbe removed as required), and addition of removable cobalt absorbers.

8.2 Burnup

One of the most important features of the CANDU reactor is the burn­up attainable from natural uranium fuel. Reactivity change as afunction of burnup is a very difficult effect to estimate for natural fuelsince it is a very fine balancc bctween the effects of burning 235u,producing 239pu, and absorption in fission products. The success ofthe Canadian system has hinged on very careful attention to neutroneconomy. Defore the successful operation of Canadian demonstrationand prototype power reactors there was widespread doubt that thepredicted burnups could be attained. Since the fuelling costs areroughly inversely proportional to the burnup, the whole conceptdepended on reaching close to these burnup levels. Many experimentswere done on which to base the estimates until a sufficient level ofconfidence was reached.

To put the burnup of 10,000 MWd/TeU in perspective, the amount ofenergy which can be derived from burning all the 235U in a tonne ofnatural U is only 6000 MWd. In fact, more than half the energy comesfrom 238U fast fissions and 239pu fission.

One of the secrets of obtaining such a high burnup from naturaluranium is the use of short fuel bundles which can be shifted in theaxial direction. If full length fuel were used, the fuel at the endswould receive a lower than average burnup whereas that at the centrewould be higher than average. Since the centre portion is by far themost important in determining the overall reactivity effect this wouldlead to a lower average burnup than in the bundle fuelling case whereit can be arranged to obtain the same average burnup from all bundlesby appropriate shifting.

Thc fuel bundles removed from the reactor after producing 10,000MWd/TeU in energy, contain about 3 g of Pu per kg of U. This Pu isvaluable either for recycling in a thermal reactor or to provide Puinventory for a fast reactor. While at present no credit is taken forthe value of this Pu, it is estimated that in the future there will be amarket for it and that this will further reduce fuelling costs for heavywater reactors.

34

8.3 Power Shape and Flux Flattening

In thermal reactors, such as existing CANDU reactors, the reactionrates are determined predominantly by the thermal neutron flux.Therefore, in what follows, flux should be taken to mean thermalneutron flux unless otherwise specified.

The spatial flux distribution in a CANDU reactor has a good deal ofstructure. It is convenient and useful to divide this structure intothree levels - macroscopic distribution, fine structure, and hyperfinestructure.

The macroscopic flux distribution is the overall flux distribution whichwould be obtained in a homogeneous reactor. Or in a heterogeneousreactor with a uniform lattice the distribution obtained from equivalentpoints (e. g. centre of fuel bundles) in each lattice cell. The macro­scopic flux distribution in a cylindrical, uniform, unreflected reactorhas the shape of a cosine in the axial direction (with the zeroes of thecosine lying just beyond the ends of the reactor) and a Bessel function(Jo> in the radial direction (with the zero lying just beyond the edge).This radial distribution is shown in Figure 16. For this macroscopicdistribution the ratio of average flux in the reactor to flux at the centreis about 0.275; the axial factor being 0.637 and the radial 0.432.Since the power of the reactor is basically limited by the centralbundle rating, there is considerable incentive to increase these factorsin order to increase the power density per unit of fuel.

1.0

X::l..JU.

w 0.5 FLATTENED>i= DISTRIBUTION« WITH REFLECTOR..JWex:

Jo[.8 rJ0

0 1 2 3

RADIUS (ARBITRARY UNITS)

Figure 16 Typical Radial Macroscopic Flux Distributions

35

In CANDU two methods are used to increase the radial factor. Thefirst is by providing a reflector, which is done essentially by notputting in some of the outer rods, and the second is by flattening theradial flux distribution in a central region. This flattening is achievedby having a lower reactivity in the central core region than the outerregion either by adjusting the burnups so that the average burnup ishigher in the central region than the outer region, or by introducingabsorbers in the central region. Figure 16 shows schematically atypical radial flux distribution in a CANDU reactor. By these meansthe radial factor is increased to 0.8 or higher.

The fine structure of the flux distribution refers to the distributionwithin a lattice cell (assuming a homogeneous rod or taking equivalentpoints within the fuel bundle such as average flux in an element). Thehyperfine structure refers to the detailed distribution within the fuelchannel itself. Figure 17 shows the hyperfine structure in moredetail. Note that the flux in the moderator is a factor of about 2higher than that in the fuel. There is also fine structure in the axialdirection, caused by the fact that short regions at the bundle ends donot contain fuel. This type of fine structure is shown schematicallyin Figure 19.

Obviously this flux distribution information is very important in fueldesign since limits are often related to the power production at somespecific position in a single element. This must be related to bundlepower through a knowledge of the detailed flux distributions.

.

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Figure 17 Fine Structure in Lattice Cell(28 cm Pitch, D20 Cooled)

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Figure 18 Hyperfine Structure in Fuel Assembly(28 cm Pitch, D20 Cooled)

38

1.7

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/1 Io 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

DISTANCE FROM END OF BUNDLE (em) -

Figure 19 Flux Peaking When Two BLW BundlesAre In Contact - Air Coolant

39

8.4 Control and Safety

There are many reasons for requiring a fairly wide range of controlof reactivity in a CANDU reactor:

(i) Power Regulation:

It must be possible to manoeuvre the power, up or down, at anytime as well as to start up and shut down.

(ii) Power and Temperature Effects:

It must be possible to compensate for changes in reactivity dueto changes in operating conditions.

(iii) Fuel Management:

Reactivity changes due to fuel burnup must be compensated aswell as changes occurring during fuelling operations.

(iv) Xenon Transients:

These are discussed below and put additional requirements onthe control system.

(v) Spatial Flux Distribution:

In large reactors there must be some scope provided forchanging the spatial flux distribution to combat xenon oscillations(see below) or other potential spatial instabilities.

To prOVide the necessary control many different methods are used inCANDU reactors; often several in a single reactor. Some of theseare briefly discussed below.

(i) On-Power Refuelling:

Fuel management provides a very slow control on reactivity.It also can be used to provide slow adjustments on the spatialpower distribution.

(ii) Boron in Moderator:

As has already been mentioned, this means is used for compen­sating the large reactivity excess associated with the initialcharge of fresh fuel.

40

(iii) Absorbers:

Solid or liquid movable absorbers are used for fast reactivitycontrol and spatial flux control.

(iv) Booster Rods:

This is the name given to rods containing fissile material whichcan be inserted or removed from the core. They are usuallyused for overcoming xenon transients (see below).

(v) Miscellaneous:

Other methods of control are also used occasionally. Forexample, in Gentilly changes in coolant flow cause changes in

reactivity and flow control is used to a certain extent.

(vi) Moderator Level:

Adjustment of moderator level has been used in some of theCANDU reactors as a fast reactivity control.

In addition to reactivity control means must be provided to insertnegative reactivity qUickly to shut down the reactor safely in the eventof some abnormal occurrence (e. g., loss of coolant). The methodsmost commonly used are fast moderator dump, shut-off (absorber)rod insertion, and fast poison injection into the moderator.

8. 5 Xenon Effects

135xe is a fission product with a relatively high yield (:.:=::: 6% from235U fission) and a very high neutron absorption cross section(:::::: 3 x 106 barns). 135Xe decays with a half life of 9.2 hours into alow cross section. 135Xe is formed (mainly) indirectly from thedecay of 135r (~ 6. 7 hour half life) which has a low neutron absorptioncross section.

These properties of 135Xe lead to some rather strange effects whichwill be summarized below.

Equilibrium Xenon:

The cross section of 135Xe is so high that in a high flux powerreactor such as CANDU most of the 135Xe is destroyed by neutron

41

capture rather than by decay. When equilibrium has beenestablished the reactivity effect of this xenon is a loss of about25-30 wk.

Xenon Transients:

When a reactor with fresh fuel is first started up to full powerthere is no 135Xe present. 1351 builds up first as operationcontinues and then 135xe as the 1351 decays. Thus the reactivityloss starts very slowly, builds up more quickly, and then levelsoff at its equilibrium value. This takes somewhat over a day ina high flux reactor. At equilibrium there is much more 135Ipresent than 135xe.

Now consider what happens if the flux is suddenly lowered. The 135Xepresent increases at first since less is being destroyed by capture andthe production from 1351 initially is constant. Later as the amount of1351 decreases (since the formation rate of 1351 is lower) the 135Xedecreases. Therefore, on a shutdown after long operation, thereactivity decreases to a minimum value (after about 10 hours) andthen slowly increases again back to the equilibrium value (after some30 hours), and finally after considerably longer back to the 135Xe-freevalue. This is called a xenon transient.

If enough reactivity is available to start up at the peak of the xenontransient, once full power has been reached, the reactivity will increasevery rapidly, since the Xe is burning out and there is very little 1351to supply more, level out, and then decrease to the value set byequilibrium xenon.

In any slow power manoeuvering 135Xe plays a strong role in deter­mining reactivity requirement~.

Xenon Oscillations:

One can see how these properties can lead to Spatial flux instabi­lity by the following argument. Suppose we start with a fixedflux distribution and equilibrium distributions of 135! and 135Xe.Now assume a flux perturbation whereby the flux at one side ofthe reactor increases a bit and at the other side decreases a bit.On the side with the slight increase in flux the 135! productionrate will go up, but this will not be immediately reflected in theXe production· rate due to the delay in the decay of the I. Thedestruction rate of Xe by neutron capture will go up due to. theflux increase. Hence initially the amount of Xe present will drop

42

and since the Xe is a heavy parasitic absorber this will causethe flux to rise further. On the other side of the reactor bysimilar arguments the amount of xenon will be going up therebydecreasing the flux still further. Therefore the flux perturbationwill initially be amplified. Eventually the Xe production rate isaffected by the I production rate and this starts to turn theprocess around and drive the fluxes in the other direction. Ifthe effects of changes in Xe on changes in flux are large enough,the result is an oscillation in flux of growing magnitude, withcycle lengths of the order of 24 hours. All types of thermalreactors are susceptible to this form of instability if they aremade large enough.